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Sdg Interacting Boson Model: Hexadecupole Degree of Freedom in Nuclear Structure T Pramana - J. Phys., Vol. 39, No. 5, November 1992, pp. 413-491. © Printed in India. sdg Interacting boson model: hexadecupole degree of freedom in nuclear structure t Y DURGA DEVI and V KRISHNA BRAHMAM KOTA Physical Research Laboratory, Ahmedabad 380009, India MS received 13 May 1992; revised 27 August 1992 Abstract. The sdg interacting boson model (sdglBM), which includes monopole (s), quadrupole (d) and hexadecupole (g) degrees of freedom, enables one to analyze hexadecupole (E4) properties of atomic nuclei. Various aspects of the model, both analytical and numerical, are reviewed emphasizing the symmetry structures involved. A large number of examples are given to provide understanding and tests, and to demonstrate the predictiveness of the sdg model. Extensions of the model to include proton-neutron degrees of freedom and fermion degrees of freedom (appropriate for odd mass nuclei) are briefly described. A comprehensive account of sdglBM analysis of all the existing data on hexadecupole observables (mainly in the rare-earth region) is presented, including l/4 (hexadecupole deformation) systematics, B(IS4; 0~s~4~+) systematics that give information about hexa'decupole component in 7-vilaratlon, E4 matrix elements involving few low-lying 4 + levels, E4 strength distributions and hexadecupole vibrational bands in deformed nuclei. Keywords. Interacting boson model; IBM; g-bosons; sdglBM; dynamical symmetries; collective modes; SU(3); coherent states; spectroscopy; mappings; 1/N method; proton-neutron IBM; F-spin; IBM-2 to IBM-1 projection; interacting boson-fermion model; scissors states; pseudo-spin; hexadeeupole deformation; E4 strength distributions; hexadecupole vibrational bands. PACS Nos 21-10; 21.60; 23.20; 27.70 Table of Contents 1. Introduction 2. sdg Interacting boson model 2.1 Dynamical symmetries 2.2 Geometric shapes 2.3 Two-nucleon transfer 2.4 Hexadecupole vibrational bands 2.5 sdg Hamiltonians 2.6 Spectroscopy in sdg space 2.7 Studies based on intrinsic states 3. Extensions of sdglBM 3.1 Proton-neutron sdglBM 3.2 sdg Interacting boson-fermion model 4. E4 Observables and sdglBM 4.1 [/4 Systematics tThe survey of literature for this review was concluded in December 1991. 413 414 Y Durga Devi and V Krishna Brahmam Kota 4.2 B(IS4; 0 GS+ ~4 ~,+) systematics 4.3 Select E4 matrix elements 4.4 E4 Strength distributions 4.5 Excited rotational bands 5. Conclusions and future outlook 1. Introduction Atomic nuclei exhibit a wide variety of collective phenomena, and their study initiated by Rainwater, Bohr and Mottelson (Rainwater 1950; Bohr 1951, 1952; Bohr and Mottelson 1953, 1975) continues to be a subject of intense investigations. The observation of enhanced (with respect to the extreme single particle model) quadrupole moments, and electric quadrupole (E2) transition strengths and the appearance of: (i)the one phonon 2 + state and the two phonon (0÷,2+,4 +) triplet states characterizing the quadrupole vibrations; (ii) the rotational spectrum characteristic of a quadrupole deformed (spheroid which is axially symmetric) object; (iii) the spectra that exhibit the so-called v-unstable motion (V is the axial asymmetry parameter), in several nuclei establish the dominance of the quadrupole collective mode in the low-lying states of medium and heavy mass nuclei; the equilibrium shapes that correspond to (i)-(iii) are spherical, spheroidal and V-soft (or ellipsoidal), respectively. Bohr and Mottelson (1975) developed the collective models (which are geometric in origin) to explain and predict the various consequences of this collective mode by seeking a multipole expansion of the nuclear surface, and treating the resulting expansion variables as collective (canonical) variables (i.e. the radius R(0,~b) is expanded as R(O,~)=Ro{I+~',~t~ Yx.(O,~)} ~>2,1~1~2 where etzu are collective variables). In the last 15 years there has been a significant amount of activity devoted to exploring other collective modes; octupole (Leander and Sheline 1984; Rohozinski 1988; Leander and Chen 1988; Kusnezov 1988, 1990), hexadecupole (see §4 ahead), scissors (Dieperink and Wenes 1985; Richter 1988, 1991) and collective states which include or-clusters (Daley and Iachello 1986), two-phonon (Borner et at 1991) and two quasi-particle states (Solov'ev and Shirikova 1981, 1989; Nesterenko et al 1986; Solov'ev 1991) etc. Figure la illustrates some of the nuclear shapes and, as an example, figure lb shows the dynamics associated with quadrupole shapes. Although the collective models were enormously successful in explaining data (Bohr and Mottelson 1975) and predicted many new phenomena in nuclei (the recent observation and exploration of super deformed bands (Twin 1988; Janssens and Khoo 1991) is a good example; super deformed shapes are prolate ellipsoids with axes ratios of 2:1:1 and they are first discovered in 15ZDy) nuclear physicists time and again tried to develop a model which in some way incorporates the microscopic (shell model) aspects of nuclei and at the same time retains the simplicity and the important features of the Bohr and Mottelson model. The outcome of this activity is the interacting boson model (IBM), which was proposed by Arima and Iachello (1975, 1978a) in 1975. Sphere Prolate Spheroid Oblate Spheroid Octupole (0,0,0) (0.3,0,0) (-0.3,0,0) (0.0.3,0) Hexadecupole Hexadecupole Quadrupole -t- Hexad{,cupoh, Quadrupole -F Hexadecupoh" (0,0,0.15) (0,0,-0.15) (0.3,0,0.15) (0.3,0,-0.15) Figure l(a). Some typical axially symmetric nuclear shapes. Corresponding to each shape, given are the parameters (fl2,fla,fl4) that define the nuclear mrface: R(O, dp)= Ro[1 + f12 Yo(O,2 d?) + f13 Y03 (0, ~b) + fla Y04 (0, ~b)]. 416 Y Durga Devi and V Krishna Brahmam Kota (b) VIBRATION ROTATION SCISSORS CLUSTERING Figure 1(b). Collective modes associated with quadrupole shape. The interacting boson model provides an algebraic description of the quadrupole collective properties of low-lying states in nuclei in terms of a system of interacting bosons. The bosons are assumed to be made up of correlated pairs of valence nucleons [as Balantekin et al (1981) put it "the correlations are so large that the bosons effectively lose the memory of beingfermion pairs---"] and they carry angular momentum f = 0 (s-boson representing pairing degree of freedom) or • = 2 (d-boson representing quadrupole degree of freedom). The bosons are allowed to interact via (1 + 2)-body forces and the boson number N (which is half the number of valence nucleons; particles or holes) is assumed to be conserved. This model is remarkably successful in explaining experimental data (Iachello and Arima 1987; Casten and Warner 1988). In the words of Feshbach (Iachello 1981) "IBM has yielded new insights into the behaviour of low-lying levels and indeed has generated a renaissance of the field of nuclear spectroscopy". One of the most important features of IBM is that it has a rich group structure. This model admits three (only three) dynamical symmetries (Arima and Iachello 1976, 1978b, 1979) denoted by the groups U(5), SU(3) and 0(6) and they describe the vibrational, axially symmetric deformed and 7-unstable nuclei, respectively. Apart from these limiting situations, the model allows for rapid analysis of experimental data in the transitional nuclei (by interpolating the limiting situations (Scholten et al 1978; Casten and Cizewski 1978, 1984; Stachel et al 1982); for example 628m, 76Os, 44Ru isotopes interpolate I-U(5), SU(3)], [SU(3), O(6)-I and [U(5), 0(6)] respectively) as the general IBM hamiltonian possesses only seven free parameters and the model-space dimensions are small (~ 100 for N ~< 15). Some of the recent reviews on the subject (Arima and Iachello 1981, 1984; Iachello and Arima 1987; Casten and Warner 1988; Dieperink and Wenes 1985; Lipas et al 1990; Iachello and Talmi 1987; Klein and Marshalek 1991; Elliott 1985; Kusnezov 1988; Kota 1987; Vervier 1987; Iachello and Van Isacker 1991; Iachello 1991) give: (i) details of the analytical results derived in the three symmetry limits and the geometric corres- pondence brought via coherent states (CS); (ii) the remarkable success in correlating observed data all across the periodic table (A/> 100); (iii) extension to proton-neutron IBM (p - n IBM or IBM-2 and the IBM where no distinction is made between protons and neutrons is often called IBM-1 or simply IBM) which gives rise to the concept of F-spin and describes the scissors I ÷ states; (iv) microscopic (shell model) basis of sd9 Interacting boson model 417 IBM; (v) IBM-3, 4 models which incorporate isospin (they allow one to study A = 20- 80 nuclei); (vi) spdf (or sdf) boson model to describe octupole collective states; (vii) extension to odd-mass nuclei via interacting boson fermion model (IBFM) where the single particle (fermion) degree of freedom is coupled to the collective (boson) degrees of freedom of the even-even core nucleus; (viii) extensions of the model to include two, three and multi-quasi particle excitations for describing high spin states, odd-odd nuclei etc. In the first two workshops (held in 1979 and 1980 respectively; Iachello 1979, 1981) that recorded the success of IBM, in his concluding remarks Feshbach raised the following questions: "the success of the interacting boson model would indicate a whole new method for looking for symmetries. As an example, are there 4 ÷ bosons and if so, what kind of symmetry is implied? In addition to the interaction among the 4 ÷ bosons, for example the 0 ÷ and 2 ÷ subspaces, what kind of broken symmetry is implied by the existence of the 4+? Comparison with the experimental data would obviously help to determine the impact of the 4 + boson .... The question we now ask has correspondingly changed. We no longer ask if the s and d boson description is adequate.
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